On the support of extremal martingale measures with given marginals: the countable case

DOI10.1017/APR.2019.16zbMath1427.60072arXiv1607.07197OpenAlexW3123428590MaRDI QIDQ5203949

Luciano Campi, Claude Martini

Publication date: 9 December 2019

Published in: Advances in Applied Probability (Search for Journal in Brave)

Abstract: We investigate the supports of extremal martingale measures with pre-specified marginals in a two-period setting. First, we establish in full generality the equivalence between the extremality of a given measure $Q$ and the denseness in $L^1(Q)$ of a suitable linear subspace, which can be seen in a financial context as the set of all semi-static trading strategies. Moreover, when the supports of both marginals are countable, we focus on the slightly stronger notion of weak exact predictable representation property (henceforth, WEP) and provide two combinatorial sufficient conditions, called "2-link property" and "full erasability", on how the points in the supports are linked to each other for granting extremality. When the support of the first marginal is a finite set, we give a necessary and sufficient condition for the WEP to hold in terms of the new concepts of $2$-net and deadlock. Finally, we study the relation between cycles and extremality.

Full work available at URL: https://arxiv.org/abs/1607.07197

zbMATH Keywords

cycleextremal measuremartingale optimal transportmodel-free pricingweak predictable representation property

Mathematics Subject Classification ID

Statistical methods; risk measures (91G70) Martingales with discrete parameter (60G42) Derivative securities (option pricing, hedging, etc.) (91G20)

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